Mathematica FindRoot exploration of parameter space

I am solving three non-linear equations in three variables (H0D,H0S and H1S) using FindRoot. In addition to the three variables of interest, there are four parameters in these equations that I would like to be able to vary. My parameters and the range in which I want to vary them are as follows:

The problem is that my non-linear system may not have any solutions for part of this parameter range. What I basically want to ask is if there is a smart way to find out exactly what part of my parameter range admits real solutions.

I could run a FindRoot inside a loop but because of non-linearity, FindRoot is very sensitive to initial conditions so frequently error messages could be because of bad initial conditions rather than absence of a solution.

Is there a way for me to find out what parameter space works, short of plugging 10^4 combinations of parameter values by hand and playing around with the initial conditions and hoping that FindRoot gives me a solution?